FA-14761 / Numerics / Open access

Bernstein to power basis: complement signed coefficient · case 01

The exact bernstein to power basis result violates the stated contract at complement signed coefficient.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The complement signed coefficient step uses math.comb(n-i,j) if j<=n-i else 0 instead of math.comb(n-i,j)*(-1)**j if j<=n-i else 0.

VERIFIED REPAIR

Use math.comb(n-i,j)*(-1)**j if j<=n-i else 0 at the complement signed coefficient step.

Unsuccessful approach: The partial repair math.comb(n-i,j)*(-1)**i if j<=n-i else 0 still violates the complement signed coefficient invariant.

Case contract

Input Bernstein coefficients b0..bn as integers; return exact ascending power coefficients of sum bi*choose(n,i)*t^i*(1-t)^(n-i).

Why this case matters

Exact discrete arithmetic with observable algorithmic state; no floating point approximation is used.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    b=x;n=len(b)-1;r=[0]*(n+1)
    for i in range(n+1):
     for j in range(n-i+1):
      k=i+j
      if k<0 or k>n:return None
      r[k]+=b[i]*(math.comb(n,i)) * (math.comb(n-i,j) if j<=n-i else 0)
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-1, -1], [-1, 0]), ([-1, 0], [-1, 1]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0], [0]), ([1], [1]), ([2], [2]), ([-1, 1], [-1, 2])], [([-1, 0], [-1, 1]), ([0, -1], [0, -1]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, 0], [2, -2]), ([2, 1], [2, -1]), ([2, 2], [2, 0]), ([-1, -1, -1], [-1, 0, 0])], [([-1, 1], [-1, 2]), ([1, -1], [1, -2]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([-1, 2, 1], [-1, 6, -4]), ([-1, 2, 2], [-1, 6, -3]), ([0, -1, -1], [0, -2, 1]), ([0, -1, 0], [0, -2, 2])], [([-1, 2], [-1, 3]), ([2, -1], [2, -3]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0, 2, 2], [0, 4, -2]), ([1, -1, -1], [1, -4, 2]), ([1, -1, 0], [1, -4, 3]), ([1, -1, 1], [1, -4, 4])], [([1, -1], [1, -2]), ([-1, 0, -1], [-1, 2, -2]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, -1, -1], [2, -6, 3]), ([2, -1, 0], [2, -6, 4]), ([2, -1, 1], [2, -6, 5]), ([2, -1, 2], [2, -6, 6])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
explicit oracle 0	`[-1, -2]`	`[-1, 0]`	Failed
explicit oracle 1	`[-1, -1]`	`[-1, 1]`	Failed
explicit oracle 2	`[-1]`	`[-1]`	Passed
explicit oracle 3	`[2, 16, 48, 64, 32]`	`[2, 0, 0, 0, 0]`	Failed
explicit oracle 4	`[0]`	`[0]`	Passed
explicit oracle 5	`[1]`	`[1]`	Passed
explicit oracle 6	`[2]`	`[2]`	Passed
explicit oracle 7	`[-1, 0]`	`[-1, 2]`	Failed

SHA-256 / 6062d2bb5ef2d9f22b8650575e61f718991e609ead66de177db32fcde3abc52e

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    b=x;n=len(b)-1;r=[0]*(n+1)
    for i in range(n+1):
     for j in range(n-i+1):
      k=i+j
      if k<0 or k>n:return None
      r[k]+=b[i]*(math.comb(n,i)) * (math.comb(n-i,j)*(-1)**i if j<=n-i else 0)
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-1, -1], [-1, 0]), ([-1, 0], [-1, 1]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0], [0]), ([1], [1]), ([2], [2]), ([-1, 1], [-1, 2])], [([-1, 0], [-1, 1]), ([0, -1], [0, -1]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, 0], [2, -2]), ([2, 1], [2, -1]), ([2, 2], [2, 0]), ([-1, -1, -1], [-1, 0, 0])], [([-1, 1], [-1, 2]), ([1, -1], [1, -2]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([-1, 2, 1], [-1, 6, -4]), ([-1, 2, 2], [-1, 6, -3]), ([0, -1, -1], [0, -2, 1]), ([0, -1, 0], [0, -2, 2])], [([-1, 2], [-1, 3]), ([2, -1], [2, -3]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0, 2, 2], [0, 4, -2]), ([1, -1, -1], [1, -4, 2]), ([1, -1, 0], [1, -4, 3]), ([1, -1, 1], [1, -4, 4])], [([1, -1], [1, -2]), ([-1, 0, -1], [-1, 2, -2]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, -1, -1], [2, -6, 3]), ([2, -1, 0], [2, -6, 4]), ([2, -1, 1], [2, -6, 5]), ([2, -1, 2], [2, -6, 6])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
explicit oracle 0	`[-1, 0]`	`[-1, 0]`	Passed
explicit oracle 1	`[-1, -1]`	`[-1, 1]`	Failed
explicit oracle 2	`[-1]`	`[-1]`	Passed
explicit oracle 3	`[2, 0, 0, 0, 0]`	`[2, 0, 0, 0, 0]`	Passed
explicit oracle 4	`[0]`	`[0]`	Passed
explicit oracle 5	`[1]`	`[1]`	Passed
explicit oracle 6	`[2]`	`[2]`	Passed
explicit oracle 7	`[-1, -2]`	`[-1, 2]`	Failed

SHA-256 / 96ca7cc5699eae1585ccca0c6b077a47021c19cb73544bbdc70b08153d7c4aad

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    b=x;n=len(b)-1;r=[0]*(n+1)
    for i in range(n+1):
     for j in range(n-i+1):
      k=i+j
      if k<0 or k>n:return None
      r[k]+=b[i]*(math.comb(n,i)) * (math.comb(n-i,j)*(-1)**j if j<=n-i else 0)
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-1, -1], [-1, 0]), ([-1, 0], [-1, 1]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0], [0]), ([1], [1]), ([2], [2]), ([-1, 1], [-1, 2])], [([-1, 0], [-1, 1]), ([0, -1], [0, -1]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, 0], [2, -2]), ([2, 1], [2, -1]), ([2, 2], [2, 0]), ([-1, -1, -1], [-1, 0, 0])], [([-1, 1], [-1, 2]), ([1, -1], [1, -2]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([-1, 2, 1], [-1, 6, -4]), ([-1, 2, 2], [-1, 6, -3]), ([0, -1, -1], [0, -2, 1]), ([0, -1, 0], [0, -2, 2])], [([-1, 2], [-1, 3]), ([2, -1], [2, -3]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0, 2, 2], [0, 4, -2]), ([1, -1, -1], [1, -4, 2]), ([1, -1, 0], [1, -4, 3]), ([1, -1, 1], [1, -4, 4])], [([1, -1], [1, -2]), ([-1, 0, -1], [-1, 2, -2]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, -1, -1], [2, -6, 3]), ([2, -1, 0], [2, -6, 4]), ([2, -1, 1], [2, -6, 5]), ([2, -1, 2], [2, -6, 6])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
explicit oracle 0	`[-1, 0]`	`[-1, 0]`	Passed
explicit oracle 1	`[-1, 1]`	`[-1, 1]`	Passed
explicit oracle 2	`[-1]`	`[-1]`	Passed
explicit oracle 3	`[2, 0, 0, 0, 0]`	`[2, 0, 0, 0, 0]`	Passed
explicit oracle 4	`[0]`	`[0]`	Passed
explicit oracle 5	`[1]`	`[1]`	Passed
explicit oracle 6	`[2]`	`[2]`	Passed
explicit oracle 7	`[-1, 2]`	`[-1, 2]`	Passed

SHA-256 / c5980ad5ed89c556d33ac718c677912154b5b3df35c214221559a2ef630bb56b

Verification & scope

A deterministic bounded teaching model. Inputs are restricted to the explicit contract; this is not a production algebra library. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:39:19.970036+00:00.

Case digest / e5849ed3efd9441474b0244c8b47714ff147142942230d30b39ca71055dd9546